Problem: Solve for $x$ : $ 6|x + 9| - 1 = -3|x + 9| + 6 $
Solution: Add $ {3|x + 9|} $ to both sides: $ \begin{eqnarray} 6|x + 9| - 1 &=& -3|x + 9| + 6 \\ \\ { + 3|x + 9|} && { + 3|x + 9|} \\ \\ 9|x + 9| - 1 &=& 6 \end{eqnarray} $ Add ${1}$ to both sides: $ \begin{eqnarray} 9|x + 9| - 1 &=& 6 \\ \\ { + 1} &=& { + 1} \\ \\ 9|x + 9| &=& 7 \end{eqnarray} $ Divide both sides by ${9}$ $ \dfrac{9|x + 9|} {{9}} = \dfrac{7} {{9}} $ Simplify: $ |x + 9| = \dfrac{7}{9}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 9 = -\dfrac{7}{9} $ or $ x + 9 = \dfrac{7}{9} $ Solve for the solution where $x + 9$ is negative: $ x + 9 = -\dfrac{7}{9} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& -\dfrac{7}{9} \\ \\ {- 9} && {- 9} \\ \\ x &=& -\dfrac{7}{9} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $9$ $ x = - \dfrac{7}{9} {- \dfrac{81}{9}} $ $ x = -\dfrac{88}{9} $ Then calculate the solution where $x + 9$ is positive: $ x + 9 = \dfrac{7}{9} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& \dfrac{7}{9} \\ \\ {- 9} && {- 9} \\ \\ x &=& \dfrac{7}{9} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $9$ $ x = \dfrac{7}{9} {- \dfrac{81}{9}} $ $ x = -\dfrac{74}{9} $ Thus, the correct answer is $x = -\dfrac{88}{9} $ or $x = -\dfrac{74}{9} $.